Coordinates and the bare insertion tests

The coordinate coefficients copied from X-A are

The density coefficients are displayed in Section sec:sheet. Retarded propagation in (eq:coordinates) reproduces the zero-data Cauchy prescription on the slab.

The copied coordinate laws are

In particular, ${\mathsf{X}}_1[K\xi ]=\xi$ on ${\mathfrak{g}}_U$. X-A's coordinate-only intertwiner also applies to compact generators supported in $\{\epsilon <t<T\}$, without requiring an interaction plateau.

Fix a real $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3)$, put $K=\mathrm{supp}f$, and

For $B^{\mu }={{\mathsf{X}}_1}^{\nu }{\partial }_{\nu }{{\mathsf{X}}_1}^{\mu }-{{\mathsf{X}}_2}^{\mu }$, the bare off-shell tests are

Write $F_c(u)=\int cu{\mathrm{d}}^{4}x$. Then ${\mathcal{O}}^{(j)}=F_{c_j}$, and ${\mathcal{O}}_{\kappa }=F_f+\kappa F_{c_1}+{\kappa }^2{F}_{c_2}$. The tests are smooth, supported in $K$, and continuously dependent on the backgrounds at fixed $\epsilon$. Their admission is copied from X-B-v3.